CPM’s Position on Acceleration of Students in Mathematics
CPM supports research-based student learning progressions that allow every student the opportunity to develop mathematical knowledge with deep understanding over time. Progressions that exacerbate gaps in content disrupt coherent trajectories in learning from grade to grade and must be avoided. CPM offers the following research and rationale for careful consideration when decisions regarding acceleration are made.
The Need to Accelerate
The reasoning behind the need to accelerate students through a mathematics curricula has both history and myth as underpinnings. For the latter half of the twentieth century, mathematics content was regularly repeated through elementary and middle school, with the percentage of topics new to the students steadily decreasing each year. Indeed, a study of 8th grade mathematics textbooks during the 1980s indicated that only about 30% of the content was new (Flanders, 1987). In parallel, myths about the necessity to rush through math courses to complete a Calculus course before applying for college have, in domino fashion, intensified a rush to complete an Algebra I course in middle school. The Common Core State Standards (CCSS) have eliminated the content redundancy of the past and current research (shared subsequently) has dispelled the myth that students must rush to complete a Calculus course before graduating high school. There is only one sound reason for acceleration: passion for mathematics.
Student Passion for Mathematics
While every student deserves the opportunity for a rich and rigorous mathematics education, a small percentage exhibit a driving passion for learning math outside of the regular classroom setting. With this elevated interest there typically is the promise of exceptional mathematical insight. Students with such promise have clearly acquired a high level of proficiency beyond grade level expectations or course-based objectives. Such competence reaches further than prodigious computational ability, a knack for memorizing procedures, or speed with routine symbolic manipulation. In particular, students with exceptional promise who also embrace challenging problems, enjoy productive struggle, seek out unique and creative solutions, take joy in communicating their ideas with others, excel at explaining complex concepts, crave a deeper understanding of a wide variety of mathematics topics, and search for all possible connections between domains. It should also be noted that exceptional mathematical promise is evenly distributed across geographic, demographic, and economic boundaries. (NCTM 2016, paraphrased).
Algebra - Cautions and Considerations
To support algebra-ready students, critical decisions about when and how to accelerate must be made. A growing body of research offers the following cautions and considerations prior to implementing any systematic acceleration.
Students who failed Algebra I in eighth grade were less likely to be on track to meet the requirements for a 4-year college by their senior year [(Howard, et al., 2015), (Nomi & Allensworth, 2014)].
If a student was unprepared to succeed in Algebra 1 in eighth grade and experienced failure (44%), the student did not benefit from simply retaking the class [(Fong, et al., 2014), (Finkelstein, et al., 2012)].
In 2002/03, the Charlotte-Mecklenburg Schools in North Carolina initiated a broad program of accelerating entry into algebra coursework. The proportion of moderately-performing students taking Algebra I in 8th grade increased from 50% to 85%, then reverted to baseline levels, in the span of just five years. Students affected by the acceleration initiative scored significantly lower on end-of-course tests in Algebra I and were either significantly less likely or no more likely to pass standard follow-up courses, such as Geometry and Algebra II, on a college-preparatory timetable (Clotfelter, et al., 2012).
Data from a large urban Kansas school district showed that of the approximately 30% of 8th graders that were accelerated into Algebra I, only one-sixth of those students enrolled in Calculus four years later. Additionally, over 80% of the accelerated students ended up repeating Algebra I, Geometry, or Algebra 2 in their senior year, or were not taking a 4th-year mathematics course.
A solid foundation in middle school mathematics followed by Algebra I better prepares a student to take an advanced course (beyond Algebra II) in high school, improving their likelihood for success in 4-year college courses (ACT, 2005). In fact, doing well in 7th grade math is highly predictive of enrollment in more advanced courses in high school (Finkelstein, et al., 2012). It is clear that middle school math plays a strong role in college and career success and cannot be rushed (NCCE report, 2013).
The National Assessment of Education Progress (NAEP) of eighth grade math scores from 2000 to 2007 shows growth in every group except those students who were accelerated (Loveless, 2008). Additionally, the scores for those in Algebra I varied widely, with the bottom 10% scoring far below grade level (Brookings Institute - Brown Center Report, 2009).
The Achieve Group (www.archive.org), an organization led by governors, business executives, and influential educational leaders committed to improving K-12 educational outcomes for all students, endorses the idea that middle school students who have completed a well-crafted sequence of compacted courses should have the opportunity to take rigorous high school mathematics. The group recommends the following: 1) taking compacted courses that include all of the Common Core State Standards as the non-compacted courses, 2) placing students into accelerated tracks too early should be avoided at all costs, 3) decisions to accelerate should be based on solid evidence of student learning, and 4) all students should be encouraged to complete four years of mathematics during high school.
Now entering its third year of implementation, San Francisco Unified School District (SFUSD) has changed its math curriculum and course sequence to reflect the Common Core State Standards and current research regarding the best practices in teaching math. In addition to rolling out a new CCSS-aligned math curriculum for K-12, SFUSD stopped offering a stand-alone Algebra I class for eighth graders. A new report from SRI International on SFUSD’s Science, Technology, Engineering and Math (STEM) Learning Initiative shows that SFUSD’s eighth grade students are ahead of their peers in other school districts when it comes to math performance. Researchers analyzed responses to a Mathematics Assessment Resource Service (MARS) task in which students developed a linear model and had to both determine the answers and explain their thinking. The report showed that SFUSD had a greater percentage of high-performing students and fewer low-performing students than the comparison group derived from a diverse group of 8,629 students from 34 different school districts (including several affluent school districts in the Bay Area). The SFUSD group included data from 599 students from 10 different SFUSD schools (SFUSD, 2016). After eliminating middle school acceleration, SFUSD experienced an 80% drop in the number of students having to repeat Algebra I, as well as a notable increase in student enrollment in STEM courses.
While it is an admirable goal to have each and every student appropriately challenged, acceleration that allows students to skip grades or courses based on simplistic cut scores has fallen short as a means to an end. Instead, a fair, objective, and transparent policy that uses multiple objective academic measures (California Mathematics Placement Act, 2015) is recommended.
Students under consideration for acceleration should understand that they will be expected to demonstrate mastery of the full range of mathematical content, undertaken in a compacted format. Engagement in enriching learning opportunities in not fulfilled by skipping content to navigate a pathway at a faster rate [(NCTM 2016), (CCSS Appendix A)].
Acceleration and the Calculus Question
Studies by the College Board and others have shown that well-prepared students who successfully complete advanced mathematics courses during high school, such as Precalculus, Calculus, and Statistics, were more likely to do well in college (Wyatt and Wilery, 2010). While it is clear that four years of rigorous mathematics coursework in grades 9 through 12 is strongly recommended in preparation for continuing education beyond high school, the widespread belief that a Calculus course is a required, gatekeeper course necessary for college admission is simply false.
In a joint statement, the Mathematics Association of America and the National Council of Teachers of Mathematics declared, “While there is an important role for calculus in secondary school, the ultimate goal of the K-12 mathematics curriculum should not be to get into and through a course in calculus by 12th grade, but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college” (MAA-NCTM position paper, 2012).
Another contributing factor promoting the “race to calculus” is the belief that an accelerated pathway will guarantee entrance to advanced mathematics courses in college. Although there has been a dramatic increase in the number of students taking a Calculus I course in high school, enrollment in a Calculus II course at the university level has remained relatively unchanged for the last two decades (Bressoud, 2004; 2009). According to a study at Rutgers University, only 5.4% of students who took Calculus I in high school followed with Calculus II in college (Rosenstein, 2014). Professors are not interested in having students race through high school courses. In an ACT survey (2012), mathematics topics from middle school and Algebra I were rated as more important for college preparation by college faculty than advanced topics. Survey respondents made it clear—it is necessary for all students to take the time needed to master the fundamentals of number, algebra, and geometry addressed in the middle school standards.
Completing Calculus I in high school has been viewed as a mandatory prerequisite for students who wish to pursue a career in science or technology. In reality, there are many paths to STEM careers, either with Calculus I in high school or with multiple alternative trajectories. Nearly half of those who went on to become scientists or engineers did not follow a prescribed high school mathematics pipeline (Cannady, et al., 2014).
Pathways for Acceleration within the CPM Curriculum
CPM Educational Program provides an option for mathematics acceleration in the middle school grades by combining the content standards of grades 6, 7, and 8 into the two school years of 6th and 7th grade, allowing a student to enroll in Algebra I in the 8th grade. This approach is a rigorous, demanding pathway, for which relatively few students are qualified; however, it does allow for a student who seems qualified but might struggle with the pathway to more easily return to the regular grade level series of courses. Additionally, no special books need to be purchased.
Alternatively, CCSS presents an acceleration pathway in Appendix A that combines the content standards of 7th and 8th grades, plus the content standards of the high school Algebra 1 course into the 7th and 8th grade years. The Appendix A pathway requires two extremely fast-paced courses, and would be challenging for even the best of students. Therefore, CPM does not support the CCSS “compacted” version of the Traditional pathway.
There are alternatives to acceleration during middle school. Students who want to take mathematics beyond Algebra II during high school (completed during the junior year) have several options including Statistics, Precalculus and Calculus I. After successfully completing all the lessons and/or standards contained within the CPM Core Connections Algebra II course, students are prepared to go directly to Calculus I. Others might choose to take CPM Precalculus or CPM Statistics. In either case, successful completion of Core Connections Algebra II makes this pathway possible. Another option is to double up on math classes, taking CPM Core Connections Geometry and Core Connections Algebra II concurrently. A final option is to take a Prealculus course the summer before taking Calculus I.
CPM agrees with the research presented earlier and believes the CCSS 6th, 7th, and 8th grade mathematics content standards are rigorous enough; no student should be pushed through these standards quickly. Much of what society knew as algebra standards before the Common Core era was moved to the 8th grade standards. Today, Algebra I is a more robust and demanding course. Many standards in a pre-CCSS era algebra course were moved to CCSS grade 8, and CCSS Algebra I now contains many topics that were previously taught in an Algebra II course. By completing a CCSS 8th grade course successfully, a student will have mastered topics associated with linear equations and functions before even beginning an algebra course; students who take Algebra I in the 9th grade will find the course content very challenging.
Every mathematics learner should be provided with a robust and challenging curriculum complete with enriching content. Every student must have the opportunity to experience the continuity of the learning progression without disruption and gain a thorough understanding of the core content before acceleration is considered. The research is clear; only a small percentage of students with exceptional mathematical promise benefit from acceleration.
Each school district must carefully examine the learning that is taking place in its K-12 mathematics classrooms and decide what to do about acceleration for its student population. CPM encourages schools to allow students to learn the CCSS mathematical content standards in the grades they are intended to be addressed. The CCSS for mathematics are designed for students to learn math through sense-making and with more depth than previously was expected. Therefore it is important to allow students the time needed to receive a comprehensive and complete mathematics education.